Energy principles

1. Lecture 6
Energy principles
Energy methods and variational principles
Print version Lecture on Theory of Elasticity and Plasticity of
Dr. D. Dinev, Department of Structural Mechanics, UACEG
6.1
Contents
1 Work and energy 1
2 Strain energy 2
3 Principle of Minimum Potential Energy 3
4 Ritz method 5 6.2
1 Work and energy
Work and energy
Work
• A material particle is moved from point A to point B by a force F
• The infinitesimal distance along the path from A to B is a displacement du
• The work dW performed by the force F is defined as
dW = F·du
• The work done is the product of the displacement and the force in the direction of the
displacement
• The total work is
W =
B
A
F·du
6.3
Work and energy
Work
1

2. • Work = Force × Displacement
6.4
Work and energy
Energy
• The energy is the capacity to do work
• It is a measure of the capacity of all forces on a body to do work
• Work is performed on a body trough a change in energy.
6.5
2 Strain energy
Strain energy
Internal work
• The work done by forces on an elastic solid is stored inside the body in the form of a strain
energy
• Consider the uniaxial tension test where we can assume that the stresses increases slowly
from zero to σ
6.6
Strain energy
Internal work
• The strain energy stored is equal to the work done on the differential element
dU =
σ
0
σd u+
∂u
∂x
dx dydz−
σ
0
σdudydz
=
σ2
2E
dxdydz
6.7
Strain energy
Internal work
• The strain energy density is the strain energy per unite volume
U =
dU
dxdydz
=
1
2
σε
6.8
2

3. Strain energy
Internal work
• The strain energy density is the shaded area under the stress-strain curve
U =
1
2
σε
6.9
Strain energy
Internal work
• General expression for the strain energy density
U =
1
2
(σxxεxx +σyyεyy +σzzεzz +2σxyεxy +2σyzεyz +2σzxεzx)
=
1
2
σijεij =
1
2
σ : ε
• The total strain energy is
Utot =
1
2 V
σ : εdV
• The potential of the applied forces is
W = −
V
f·udV +
S
t·udS
6.10
3 Principle of Minimum Potential Energy
Principle of Minimum Potential Energy
TPE functional
• Total potential energy of the body
ΠTPE = Utot +W =
1
2 V
σ : εdV −
V
f·udV −
S
t·udS
6.11
Principle of Minimum Potential Energy
TPE functional
• The principle states that: The body is in equilibrium if there is an admissible displacement
field u that makes the total potential energy a minimum
δΠTPE(u) = 0
where δΠTPE is a variation of the functional ΠTPE(u)
• The admissible displacement field is the one that satisfy the displacement BCs.
Note
The variational operator δ is much like the differential operator d except that it operates with
respect to the dependent variable u rather than the independent variable x
6.12
3

4. Principle of Minimum Potential Energy
0.5 0.5 1.0 1.5 2.0
u, u
500
500
1000
W, U,
TPE functional
• Local minimum of the TPE functional
6.13
Principle of Minimum Potential Energy
A gentle touch to the variational analysis
• The condition for minimum of a functional I(u) is
δI =
∂I
∂u
δu = 0
• Almost the same as the condition for a minimum of a function u(x)
du =
∂u
∂x
dx = 0
6.14
Principle of Minimum Potential Energy
A gentle touch to the variational analysis
• Consider a functional
I(u) =
b
a
F x,u(x),u (x) dx
• The minimum condition is
δI(u) =
b
a
∂F
∂x
δx+
∂F
∂u
δu+
∂F
∂u
δu dx = 0
• The integral can be manipulated to get the expression for the variation of u (integration by
parts of the 3-rd addend)
6.15
Principle of Minimum Potential Energy
A gentle touch to the variational analysis
• The variation of the functional is
δI(u) =
b
a
∂F
∂u
−
∂
∂x
∂F
∂u
δudx
• The non-trivial solution gives
∂F
∂u
−
∂
∂x
∂F
∂u
= 0
• The above expression is called Euler equation
6.16
4

5. Principle of Minimum Potential Energy
A gentle touch to the variational analysis
• A more general 2D case is given by
I(u,v) =
A
F (x,y,u,v,u,x,v,x,...,v,yy)
• The Euler equations are
∂F
∂u
−
∂
∂x
∂F
∂u,x
−
∂
∂y
∂F
∂u,y
+
∂2
∂x2
∂F
∂u,xx
+
∂2
∂x∂y
∂F
∂u,xy
+
∂2
∂y2
∂F
∂u,yy
= 0
∂F
∂u
−
∂
∂x
∂F
∂u,x
−
∂
∂y
∂F
∂u,y
+
∂2
∂x2
∂F
∂u,xx
+
∂2
∂x∂y
∂F
∂u,xy
+
∂2
∂y2
∂F
∂u,yy
= 0
6.17
Principle of Minimum Potential Energy
Example
• Consider a cantilever beam with length of and subjected to uniform load q. Using the
principle of a minimum of TPE work out the equilibrium equation
6.18
4 Ritz method
Ritz method
Approximate solution
• A lot of problems in elasticity there is no analytical solution of the field equations
• For such cases approximate solution schemes have been developed based on the varia-
tional formulation of the problem (i.e. the principle of minimum potential energy)
• The Ritz method is based on the idea of constructing a series of trial approximating func-
tions that satisfy the essential (displacement) BCs but not differential equations exactly
Note
• Since the TPE functional includes the force BCs, it is require the trial solution satisfies
only the displacement BCs!
6.19
Ritz method
Approximate solution
• Walter Ritz (1878-1909)
6.20
5

6. Ritz method
Approximate solution
• The original paper
6.21
Ritz method
Approximate solution
• The displacement can be expressed as
u = u0 +a1u1 +a2u2 +...+anun = u0 +
n
∑
i=1
aiui
v = v0 +b1v1 +b2v2 +...+bnvn = v0 +
n
∑
i=1
bivi
w = w0 +c1w1 +c2w2 +...+cnwn = w0 +
n
∑
i=1
ciwi
• The terms of u0, v0 and w0 are chosen to satisfy any non-homogeneous displacement BCs
and ui, vi and wi satisfy the corresponding homogeneous BCs
• These forms are not required to satisfy the stress BCs
6.22
Ritz method
Approximate solution
• These trial functions are chosen from the some combinations of elementary functions
(polynomials, trigonometric or hyperbolic forms)
• The unknown coefficients ai, bi and ci are to be determined so as to minimize the TPE
functional of the problem
• Thus we approximately satisfy the variational formulation of the problem
• Using this approximation the TPE functional will be a function of these unknown coeffi-
cients
ΠTPE = ΠTPE(ai,bi,ci)
6.23
Ritz method
Approximate solution
• The minimizing condition ca be expressed as a series of
∂ΠTPE
∂ai
= 0,
∂ΠTPE
∂bi
= 0,
∂ΠTPE
∂ci
= 0
• This set forms a system of 3n equations which gives ai, bi and ci
6

7. • Under suitable conditions on the choice of trial functions (completeness) the approxima-
tion will improve as the number of included terms is increased
• When the approximate displacement solution is obtained the strains and stresses can be
calculated from the appropriate field equations
Note
The method is suitable to apply at problems involving one or two displacements (bars, beams,
plates and shells)
6.24
Ritz method
Example
• Consider a bending of simply supported beam of length carrying a uniform load q
6.25
Ritz method
The End
• Any questions, opinions, discussions?
6.26
7